Zbigniew Błocki (2004)
Annales Polonici Mathematici
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We give upper and lower bounds for constants appearing in the L²-estimates for the ∂̅-operator due to Donnelly-Fefferman and Berndtsson.
Yves Capdeboscq, Michael S. Vogelius (2010)
ESAIM: Mathematical Modelling and Numerical Analysis
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We recently derived a very general representation formula for the boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction ( Capdeboscq and Vogelius (2003)). In this paper we show how this representation formula may be used to obtain very accurate estimates for the size of the inhomogeneities in terms of multiple boundary measurements. As demonstrated by our computational experiments, these estimates are significantly better than previously...
Sergey Repin (2002)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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The paper is concerned with deriving functionals that give upper bounds of the difference between the exact solution of the initial-boundary value problem for the heat equation and any admissible function from the functional class naturally associated with this problem. These bounds are given by nonegative functionals called deviation majorants, which vanish only if the function and exact solution coincide. The deviation majorants pose a new type of a posteriori estimates that can be...
Yves Capdeboscq, Michael S. Vogelius (2003)
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
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We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.
Peter Wall (2000)
Applications of Mathematics
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In this paper we derive lower bounds and upper bounds on the effective properties for nonlinear heterogeneous systems. The key result to obtain these bounds is to derive a variational principle, which generalizes the variational principle by P. Ponte Castaneda from 1992. In general, when the Ponte Castaneda variational principle is used one only gets either a lower or an upper bound depending on the growth conditions. In this paper we overcome this problem by using our new variational...